English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the problem of robustly estimating the mean or location parameterwithout moment assumptions. We show that for a large class of symmetricdistributions, the same error as in the Gaussian setting can be achievedefficiently. The distributions we study include products of arbitrary symmetricone-dimensional distributions, such as product Cauchy distributions, as well aselliptical distributions.  For product distributions and elliptical distributions with known scatter(covariance) matrix, we show that given an $\varepsilon$-corrupted sample, wecan with probability at least $1-\delta$ estimate its location up to error$O(\varepsilon \sqrt{\log(1/\varepsilon)})$ using $\tfrac{d\log(d) +\log(1/\delta)}{\varepsilon^2 \log(1/\varepsilon)}$ samples. This resultmatches the best-known guarantees for the Gaussian distribution and known SQlower bounds (up to the $\log(d)$ factor). For elliptical distributions withunknown scatter (covariance) matrix, we propose a sequence of efficientalgorithms that approaches this optimal error. Specifically, for every $k \in\mathbb{N}$, we design an estimator using time and samples $\tilde{O}({d^k})$achieving error $O(\varepsilon^{1-\frac{1}{2k}})$. This matches the error andrunning time guarantees when assuming certifiably bounded moments of order upto $k$. For unknown covariance, such error bounds of $o(\sqrt{\varepsilon})$are not even known for (general) sub-Gaussian distributions.  Our algorithms are based on a generalization of the well-known filteringtechnique. We show how this machinery can be combined with Huber-loss-basedtechniques to work with projections of the noise that behave more nicely thanthe initial noise. Moreover, we show how SoS proofs can be used to obtainalgorithmic guarantees even for distributions without a first moment. Webelieve that this approach may find other applications in future works.

Robust Mean Estimation Without Moments for Symmetric Distributions